Oscillation conditions for difference equations with several variable arguments

George E. Chatzarakis; Takaŝi Kusano; Ioannis P. Stavroulakis

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 3, page 291-311
  • ISSN: 0862-7959

Abstract

top
Consider the difference equation Δ x ( n ) + ∑ i = 1 m p i ( n ) x ( τ i ( n ) ) = 0 , n ≥ 0 ∇ x ( n ) - ∑ i = 1 m p i ( n ) x ( σ i ( n ) ) = 0 , n ≥ 1 , where ( p i ( n ) ) , 1 ≤ i ≤ m are sequences of nonnegative real numbers, τ i ( n ) [ σ i ( n ) ], 1 ≤ i ≤ m are general retarded (advanced) arguments and Δ [ ∇ ] denotes the forward (backward) difference operator Δ x ( n ) = x ( n + 1 ) - x ( n ) [ ∇ x ( n ) = x ( n ) - x ( n - 1 ) ]. New oscillation criteria are established when the well-known oscillation conditions lim sup n → ∞ ∑ i = 1 m ∑ j = τ ( n ) n p i ( j ) > 1 lim sup n → ∞ ∑ i = 1 m ∑ j = n σ ( n ) p i ( j ) > 1 and lim inf n → ∞ ∑ i = 1 m ∑ j = τ i ( n ) n - 1 p i ( j ) > 1 e lim inf n → ∞ ∑ i = 1 m ∑ j = n + 1 σ i ( n ) p i ( j ) > 1 e are not satisfied. Here τ ( n ) = max 1 ≤ i ≤ m τ i ( n ) [ σ ( n ) = min 1 ≤ i ≤ m σ i ( n ) ] . The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

How to cite

top

Chatzarakis, George E., Kusano, Takaŝi, and Stavroulakis, Ioannis P.. "Oscillation conditions for difference equations with several variable arguments." Mathematica Bohemica 140.3 (2015): 291-311. <http://eudml.org/doc/271621>.

@article{Chatzarakis2015,
abstract = {Consider the difference equation \[ \Delta x(n)+\sum \_\{i=1\}^\{m\}p\_\{i\}(n)x(\tau \_\{i\}(n))=0,\quad n\ge 0\quad \bigg [\nabla x(n)-\sum \_\{i=1\}^\{m\}p\_\{i\}(n)x(\sigma \_\{i\}(n))=0,\quad n\ge 1\bigg ], \] where $(p_\{i\}(n))$, $1\le i\le m$ are sequences of nonnegative real numbers, $\tau _\{i\}(n)$ [$\sigma _\{i\}(n)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta $ [$\nabla $] denotes the forward (backward) difference operator $\Delta x(n)=x(n+1)-x(n)$ [$\nabla x(n)=x(n)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions \[ \limsup \_\{n\rightarrow \infty \}\sum \_\{i=1\}^\{m\}\sum \_\{j=\tau (n)\}^\{n\}p\_\{i\}(j)>1 \quad \bigg [\limsup \_\{n\rightarrow \infty \}\sum \_\{i=1\}^\{m\}\sum \_\{j=n\}^\{\sigma (n)\}p\_\{i\}(j)>1\bigg ] \] and \[ \liminf \_\{n\rightarrow \infty \}\sum \_\{i=1\}^\{m\}\sum \_\{j=\tau \_\{i\}(n)\}^\{n-1\}p\_\{i\}(j)>\frac\{1\}\{\rm e\} \quad \bigg [\liminf \_\{n\rightarrow \infty \}\sum \_\{i=1\}^\{m\}\sum \_\{j=n+1\}^\{\sigma \_\{i\}(n)\}p\_\{i\}(j)>\frac\{1\}\{\rm e\}\bigg ] \] are not satisfied. Here $\tau (n)=\max _\{1\le i\le m\}\tau _\{i\}(n)$$[ \sigma (n)=\min _\{1\le i\le m\}\sigma _\{i\}(n) ]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.},
author = {Chatzarakis, George E., Kusano, Takaŝi, Stavroulakis, Ioannis P.},
journal = {Mathematica Bohemica},
keywords = {difference equation; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution},
language = {eng},
number = {3},
pages = {291-311},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Oscillation conditions for difference equations with several variable arguments},
url = {http://eudml.org/doc/271621},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Chatzarakis, George E.
AU - Kusano, Takaŝi
AU - Stavroulakis, Ioannis P.
TI - Oscillation conditions for difference equations with several variable arguments
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 3
SP - 291
EP - 311
AB - Consider the difference equation \[ \Delta x(n)+\sum _{i=1}^{m}p_{i}(n)x(\tau _{i}(n))=0,\quad n\ge 0\quad \bigg [\nabla x(n)-\sum _{i=1}^{m}p_{i}(n)x(\sigma _{i}(n))=0,\quad n\ge 1\bigg ], \] where $(p_{i}(n))$, $1\le i\le m$ are sequences of nonnegative real numbers, $\tau _{i}(n)$ [$\sigma _{i}(n)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta $ [$\nabla $] denotes the forward (backward) difference operator $\Delta x(n)=x(n+1)-x(n)$ [$\nabla x(n)=x(n)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions \[ \limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau (n)}^{n}p_{i}(j)>1 \quad \bigg [\limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n}^{\sigma (n)}p_{i}(j)>1\bigg ] \] and \[ \liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau _{i}(n)}^{n-1}p_{i}(j)>\frac{1}{\rm e} \quad \bigg [\liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n+1}^{\sigma _{i}(n)}p_{i}(j)>\frac{1}{\rm e}\bigg ] \] are not satisfied. Here $\tau (n)=\max _{1\le i\le m}\tau _{i}(n)$$[ \sigma (n)=\min _{1\le i\le m}\sigma _{i}(n) ]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
LA - eng
KW - difference equation; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution
UR - http://eudml.org/doc/271621
ER -

References

top
  1. Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D., Discrete Oscillation Theory, Hindawi Publishing Corporation New York (2005). (2005) Zbl1084.39001MR2179948
  2. Baštinec, J., Berezansky, L., Diblík, J., Šmarda, Z., A final result on the oscillation of solutions of the linear discrete delayed equation Δ x ( n ) = - p ( n ) x ( n - k ) with a positive coefficient, Abstr. Appl. Anal. 2011 (2011), Article No. 586328, 28 pages. (2011) Zbl1223.39008MR2824906
  3. Baštinec, J., Diblík, J., 10.1016/j.na.2005.01.007, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods (electronic only) 63 (2005), e2145--e2151. (2005) Zbl1224.39002DOI10.1016/j.na.2005.01.007
  4. Baštinec, J., Diblík, J., Šmarda, Z., 10.1080/10236190902718026, J. Difference Equ. Appl. 16 (2010), 1047-1056. (2010) Zbl1207.39014MR2722821DOI10.1080/10236190902718026
  5. Berezansky, L., Braverman, E., On existence of positive solutions for linear difference equations with several delays, Adv. Dyn. Syst. Appl. 1 (2006), 29-47. (2006) Zbl1124.39002MR2287633
  6. Chatzarakis, G. E., Koplatadze, R., Stavroulakis, I. P., 10.2140/pjm.2008.235.15, Pac. J. Math. 235 (2008), 15-33. (2008) Zbl1153.39010MR2379767DOI10.2140/pjm.2008.235.15
  7. Chatzarakis, G. E., Manojlovic, J., Pinelas, S., Stavroulakis, I. P., Oscillation criteria of difference equations with several deviating arguments, Yokohama Math. J. 60 (2014), 13-31. (2014) Zbl1318.39011MR3328615
  8. Chatzarakis, G. E., Philos, C. G., Stavroulakis, I. P., 10.4171/PM/1853, Port. Math. (N.S.) 66 (2009), 513-533. (2009) Zbl1186.39010MR2567680DOI10.4171/PM/1853
  9. Chatzarakis, G. E., Pinelas, S., Stavroulakis, I. P., 10.1007/s00010-013-0238-2, Aequationes Math. 88 (2014), 105-123. (2014) Zbl1306.39007MR3250787DOI10.1007/s00010-013-0238-2
  10. Erbe, L. H., Zhang, B. G., Oscillation of discrete analogues of delay equations, Differ. Integral Equ. 2 (1989), 300-309. (1989) Zbl0723.39004MR0983682
  11. Fukagai, N., Kusano, T., Oscillation theory of first order functional-differential equations with deviating arguments, Ann. Mat. Pura Appl. (4) 136 (1984), 95-117. (1984) Zbl0552.34062MR0765918
  12. Grammatikopoulos, M. K., Koplatadze, R., Stavroulakis, I. P., 10.1515/GMJ.2003.63, Georgian Math. J. 10 (2003), 63-76. (2003) Zbl1051.34051MR1990688DOI10.1515/GMJ.2003.63
  13. Győri, I., Ladas, G., Oscillation Theory of Delay Differential Equations: With Applications, Oxford Mathematical Monographs Clarendon Press, Oxford (1991). (1991) Zbl0780.34048MR1168471
  14. Lakshmikantham, V., Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering 181 Academic Press, Boston (1988). (1988) Zbl0683.39001MR0939611
  15. Li, X., Zhu, D., Oscillation of advanced difference equations with variable coefficients, Ann. Differ. Equations 18 (2002), 254-263. (2002) Zbl1010.39001MR1940383
  16. Luo, X. N., Zhou, Y., Li, C. F., Oscillation of a nonlinear difference equation with several delays, Math. Bohem. 128 (2003), 309-317. (2003) Zbl1055.39015MR2012607
  17. Stavroulakis, I. P., 10.1016/j.amc.2013.10.041, Appl. Math. Comput. 226 (2014), 661-672. (2014) Zbl1354.34120MR3144341DOI10.1016/j.amc.2013.10.041
  18. Tang, X. H., Yu, J. S., 10.14492/hokmj/1350912965, Hokkaido Math. J. 29 (2000), 213-228. (2000) Zbl0958.39015MR1745511DOI10.14492/hokmj/1350912965
  19. Tang, X. H., Yu, J. S., 10.1016/S0898-1221(99)00083-8, Comput. Math. Appl. 37 (1999), 11-20. (1999) Zbl0937.39012MR1688201DOI10.1016/S0898-1221(99)00083-8
  20. Tang, X. H., Zhang, R. Y., 10.1016/S0898-1221(01)00243-7, Comput. Math. Appl. 42 (2001), 1319-1330. (2001) Zbl1002.39022MR1861531DOI10.1016/S0898-1221(01)00243-7
  21. Yan, W., Meng, Q., Yan, J., Oscillation criteria for difference equation of variable delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal. 13A (2006), 641-647. (2006) MR2219618

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.